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On a diffusive predator-prey model with nonlinear harvesting
Coexistence and asymptotic stability in stage-structured predator-prey models
1. | Department of Mathematics and Statistics, UNC Wilmington, Wilmington, NC 28403 |
2. | Mathematics and Statistics Department, University of North Carolina Wilmington, Wilmington, NC 28403-5970, United States, United States |
References:
[1] |
P. A. Abrams and C. Quince, The impact of mortality rate on predator population size and stability in systems with stage-structured prey, Theoretical Population Biology, 68 (2005), 253-266. |
[2] |
B. Buonomo, D. Lacitignola and S. Rionero, Effect of prey growth and predator cannibalism rate on the stability of a structured population model, Nonlinear Analysis: Real World Applications, 11 (2010), 1170-1181.
doi: 10.1016/j.nonrwa.2009.01.053. |
[3] |
L. Cai and X. Song, Permanence and stability of a predator-prey system with stage structure for predator, Journal of Computational and Applied Mathematics, 201 (2007), 356-366.
doi: 10.1016/j.cam.2005.12.035. |
[4] |
R. S. Cantrell and C. Cosner, Permanence in ecological systems with spatial heterogeneity, Proc. Roy. Soc. Edinburgh, 123 (1993), 533-559.
doi: 10.1017/S0308210500025877. |
[5] |
F. Chen, Permanence of periodic Holling type predator-prey system with stage structure for prey, Applied Mathematics and Computation, 182 (2006), 1849-1860.
doi: 10.1016/j.amc.2006.06.024. |
[6] |
D. Cooke and J. A. Leon, Stability of population growth determined by 2x2 Leslie Matrix with density dependent elements, Biometrics, 32 (1976), 435-442.
doi: 10.2307/2529512. |
[7] |
J. M. Cushing and M. Saleem, A predator prey model with age structure, Journal of Mathematical Biology, 14 (1982), 231-250.
doi: 10.1007/BF01832847. |
[8] |
M. Fan and Y. Kuang, Dynamics of a nonautonomous predator-prey system with the Beddington-DeAngelis functional response, J. Math. Anal. Appl., 295 (2004), 15-39.
doi: 10.1016/j.jmaa.2004.02.038. |
[9] |
W. Feng, Coexistence, stability, and limiting behavior in a one-predator-two-prey model, J. Math. Anal. Appl., 179 (1993), 592-609.
doi: 10.1006/jmaa.1993.1371. |
[10] |
W. Feng, Permanence effect in a three-species food chain model, Applicable Analysis, 54 (1994), 195-209.
doi: 10.1080/00036819408840277. |
[11] |
W. Feng and X. Lu, Harmless delays for permanence in a class of population models with diffusion effects, J. Math. Anal. Appl., 206 (1997), 547-566.
doi: 10.1006/jmaa.1997.5265. |
[12] |
W. Feng and X. Lu, Some coexistence and extinction results in a three species ecological model, Diff. Integ. Eq.s, 8 (1995), 617-626. |
[13] |
W. Feng, C. V. Pao and X. Lu, Global attractors of reaction-diffusion systems modeling food chain populations with delays, Commun. Pure Appl. Anal., 10 (2011), 1463-1478.
doi: 10.3934/cpaa.2011.10.1463. |
[14] |
A. Hastings, Age-dependent predation is not a simple process, I: Continuous time models, Theorertical Population Biology, 23 (1983), 347-362.
doi: 10.1016/0040-5809(83)90023-0. |
[15] |
A. Leung, Limiting behaviour for a prey-predator model with diffusion and crowding effects, J. Math. Biol., 6 (1978), 87-93.
doi: 10.1007/BF02478520. |
[16] |
B. C. Longstaff, The dynamics of collembolan population growth, Canadian Journal of Zoology, 55 (1977), 314-324.
doi: 10.1139/z77-043. |
[17] |
C. Lu, W. Feng and X. Lu, Long-term survival in a 3-species ecological system, Dynam. Contin. Discrete Impuls. Systems, 3 (1997), 199-213. |
[18] |
G. A. Oster, Internal variables in population dynamics, In Levin, S.A. (Ed.), Lectures on Mathematics in the Life Sciences, 8 (1976), American Mathematical Society, Providence, RI, 37-68. |
[19] |
C. V. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum Press, N.Y., 1992. |
[20] |
R. H. Smith and R. Mead, Age structure and stability in models of prey-predator systems, Theoretical Population Biology, 6 (1974), 308-322.
doi: 10.1016/0040-5809(74)90014-8. |
[21] |
C. Sun, Y. Lin and M. Han, Dynamic analysis for a stage-structure competitive system with mature population harvesting, Chaos, Solitons & Fractals, 31 (2007), 380-390.
doi: 10.1016/j.chaos.2005.09.057. |
[22] |
H. Wang, J. D. Nagy, O. Gilg and Y. Kuang, The roles of predator maturation delay and functional response in determining the periodicity of predator-prey cycles, Mathematical Biosciences, 221 (2009), 1-10.
doi: 10.1016/j.mbs.2009.06.004. |
show all references
References:
[1] |
P. A. Abrams and C. Quince, The impact of mortality rate on predator population size and stability in systems with stage-structured prey, Theoretical Population Biology, 68 (2005), 253-266. |
[2] |
B. Buonomo, D. Lacitignola and S. Rionero, Effect of prey growth and predator cannibalism rate on the stability of a structured population model, Nonlinear Analysis: Real World Applications, 11 (2010), 1170-1181.
doi: 10.1016/j.nonrwa.2009.01.053. |
[3] |
L. Cai and X. Song, Permanence and stability of a predator-prey system with stage structure for predator, Journal of Computational and Applied Mathematics, 201 (2007), 356-366.
doi: 10.1016/j.cam.2005.12.035. |
[4] |
R. S. Cantrell and C. Cosner, Permanence in ecological systems with spatial heterogeneity, Proc. Roy. Soc. Edinburgh, 123 (1993), 533-559.
doi: 10.1017/S0308210500025877. |
[5] |
F. Chen, Permanence of periodic Holling type predator-prey system with stage structure for prey, Applied Mathematics and Computation, 182 (2006), 1849-1860.
doi: 10.1016/j.amc.2006.06.024. |
[6] |
D. Cooke and J. A. Leon, Stability of population growth determined by 2x2 Leslie Matrix with density dependent elements, Biometrics, 32 (1976), 435-442.
doi: 10.2307/2529512. |
[7] |
J. M. Cushing and M. Saleem, A predator prey model with age structure, Journal of Mathematical Biology, 14 (1982), 231-250.
doi: 10.1007/BF01832847. |
[8] |
M. Fan and Y. Kuang, Dynamics of a nonautonomous predator-prey system with the Beddington-DeAngelis functional response, J. Math. Anal. Appl., 295 (2004), 15-39.
doi: 10.1016/j.jmaa.2004.02.038. |
[9] |
W. Feng, Coexistence, stability, and limiting behavior in a one-predator-two-prey model, J. Math. Anal. Appl., 179 (1993), 592-609.
doi: 10.1006/jmaa.1993.1371. |
[10] |
W. Feng, Permanence effect in a three-species food chain model, Applicable Analysis, 54 (1994), 195-209.
doi: 10.1080/00036819408840277. |
[11] |
W. Feng and X. Lu, Harmless delays for permanence in a class of population models with diffusion effects, J. Math. Anal. Appl., 206 (1997), 547-566.
doi: 10.1006/jmaa.1997.5265. |
[12] |
W. Feng and X. Lu, Some coexistence and extinction results in a three species ecological model, Diff. Integ. Eq.s, 8 (1995), 617-626. |
[13] |
W. Feng, C. V. Pao and X. Lu, Global attractors of reaction-diffusion systems modeling food chain populations with delays, Commun. Pure Appl. Anal., 10 (2011), 1463-1478.
doi: 10.3934/cpaa.2011.10.1463. |
[14] |
A. Hastings, Age-dependent predation is not a simple process, I: Continuous time models, Theorertical Population Biology, 23 (1983), 347-362.
doi: 10.1016/0040-5809(83)90023-0. |
[15] |
A. Leung, Limiting behaviour for a prey-predator model with diffusion and crowding effects, J. Math. Biol., 6 (1978), 87-93.
doi: 10.1007/BF02478520. |
[16] |
B. C. Longstaff, The dynamics of collembolan population growth, Canadian Journal of Zoology, 55 (1977), 314-324.
doi: 10.1139/z77-043. |
[17] |
C. Lu, W. Feng and X. Lu, Long-term survival in a 3-species ecological system, Dynam. Contin. Discrete Impuls. Systems, 3 (1997), 199-213. |
[18] |
G. A. Oster, Internal variables in population dynamics, In Levin, S.A. (Ed.), Lectures on Mathematics in the Life Sciences, 8 (1976), American Mathematical Society, Providence, RI, 37-68. |
[19] |
C. V. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum Press, N.Y., 1992. |
[20] |
R. H. Smith and R. Mead, Age structure and stability in models of prey-predator systems, Theoretical Population Biology, 6 (1974), 308-322.
doi: 10.1016/0040-5809(74)90014-8. |
[21] |
C. Sun, Y. Lin and M. Han, Dynamic analysis for a stage-structure competitive system with mature population harvesting, Chaos, Solitons & Fractals, 31 (2007), 380-390.
doi: 10.1016/j.chaos.2005.09.057. |
[22] |
H. Wang, J. D. Nagy, O. Gilg and Y. Kuang, The roles of predator maturation delay and functional response in determining the periodicity of predator-prey cycles, Mathematical Biosciences, 221 (2009), 1-10.
doi: 10.1016/j.mbs.2009.06.004. |
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